The above cohomology groups have been computed in [1]} (and by alternative means
in [2]}, [3]}). They are isomorphic to the Gelfand-Fuchs
cohomology of the Lie algebra \( \mathfrak {a}_n\)  of formal vector fields
on \({\mathbb {R}}^n\) , resp. to its relative versions, which in turn are isomorphic to
the cohomology of truncated Weil algebras (cf. [4]}, [5]}), as follows:
\( HP^\bullet ({\mathcal {H}}_n; {\mathbb {C}}_\delta ) \cong H_{\rm cont}^{\bullet }(\mathfrak {a}_n; {\mathbb {C}})\cong H^\bullet ({W}(\mathfrak {g}\mathfrak {l}_n)_n) ,\) 
